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AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL PRECONDITIONING II: STABILIZING HIERARCHICAL BASIS
 

Summary: AN ODYSSEY INTO LOCAL REFINEMENT AND MULTILEVEL
PRECONDITIONING II: STABILIZING HIERARCHICAL BASIS
METHODS
BURAK AKSOYLU AND MICHAEL HOLST
Abstract. In this article, we examine the wavelet modified (or stabilized) hierarchical basis
(WHB) methods of Vassilevski and Wang, and extend their original quasiuniformity-based framework
and results to local 2D and 3D red-green refinement procedures. The concept of a stable Riesz
basis plays a critical role in the original work on WHB, and in the design of efficient multilevel
preconditioners in general. We carefully examine the impact of local mesh refinement on Riesz bases
and matrix conditioning. In the analysis of WHB methods, a critical first step is to establish that the
BPX preconditioner is optimal for the refinement procedures under consideration. Therefore, the first
article in this series was devoted to extending the results of Dahmen and Kunoth on the optimality
of BPX for 2D local red-green refinement to 3D local red-green refinement procedures introduced by
Bornemann-Erdmann-Kornhuber (BEK). These results from the first article, together with the local
refinement extension of the WHB analysis framework presented here, allow us to establish optimality
of the WHB preconditioner on locally refined meshes in both 2D and 3D. In particular, with the
minimal smoothness assumption that the PDE coefficients are in L, we establish optimality for the
additive WHB preconditioner on locally refined 2D and 3D meshes. An interesting implication of the
optimality of WHB preconditioner is the a priori H1-stability of the L2-projection. The existing a
posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot

  

Source: Aksoylu, Burak - Center for Computation and Technology & Department of Mathematics, Louisiana State University
Holst, Michael J. - Department of Mathematics, University of California at San Diego

 

Collections: Computer Technologies and Information Sciences; Mathematics